X Variables

Week 10

Evie Zhang

Old Dominion University

Topics

Exam 2
Spurious Regression
Cointegration

Exam 2

X predicts Y

Thus far, we have only used past $Y$ values to forecast future $Y$ values.

Of course, other things (besides $e$) can effect $Y$

In forecasting, we call these leading indicators

Example: housing starts, building permits, Starbucks (?)

Yield Curve

Interest rates denote the risk of an assets.

higher rates $\implies$ greater risk

Yield Curve

Length on the X axis, Yield on the Y axis.
Should be an upward slope.

An inverted curve has correctly predicted the past 7 recessions.

Yield Curve

Regressing Y on X

\[\% \Delta PCE_t = f(\% \Delta \text{Income}_t, \% \Delta \text{Savings}_t, \Delta \text{U-Rate}_t) + e_t\]

Code

macro <- read.csv("../data/macro.csv")
macro$DATE <- ymd(macro$DATE)

Regressing Y on X

Code

macrots <- ts(macro, start = c(min(year(macro$DATE)), 1, 1))
acf(macrots[,2:4])

Regressing Y on X

Regressing Y on X

To forecast using this type of model, you will need “future” values of your explanatory variables.

This is easy for some thing, and difficult for other
Vector Autoregression (VAR)
Impulse Response Functions (IRF)

Regressing Y on X

OK, so estimate $Y_t = f(X_t, e_t)$!

\[\Delta GDP_t = \alpha + \beta \Delta GDP_{t-1} + \gamma \text{Spread}_{t-1} + e_t\]

But first … what about stationarity?

Spurious Regression

Suppose you have two independent time series $y_t$ and $x_t$.

If you regress $y_t$ on $x_t$ ($y_t = \alpha + \beta x_t + e_t$), what should $\hat{\beta}$ be?

Spurious Regression

Code

tz <- c()

for(j in 1:1000){
  
  x <- rnorm(120)
  y <- rnorm(120)
  
  tz[j] <- summary(lm(y ~ x))$coefficients[2,3]
}

plot(table(round(tz, 1)),
     ylab = "Frequency",
     xlab = paste0(round(100*mean(abs(tz) > 1.96)),
                   "% Significant"))
abline(v = 0, col = "tomato")

Spurious Regression

Code

for(j in 1:1000){
  
  x <- rep(0, 120)
  y <- rep(0, 120)
  
  for(i in 2:120){
    
    x[i] <- x[i-1] + rnorm(1)
    y[i] <- y[i-1] + rnorm(1)
  }
  
  tz[j] <- summary(lm(y ~ x))$coefficients[2,3]
}

plot(table(round(tz, 0)),
     ylab = "Frequency",
     xlab = paste0(round(100*mean(abs(tz) > 1.96)),
                   "% Significant"))
abline(v = 0, col = "tomato")

Spurious Regression

This happens when you regress two random walks on one another.

Code

par(mfrow = c(1, 2))
plot(x, type = "l", col = alpha("tomato", .6),
     xlab = "Time", ylab = "",
     ylim = range(x, y))
lines(y, col = alpha("dodgerblue", .6))
plot(x, y, pch = 19, col = alpha("black", .6),
     xlab = "X", ylab = "Y")
abline(lm(y ~ x), lty = 2)

Spurious Regression

Code

reg1 <- lm(y ~ x)
t <- 2:length(y)
reg2 <- lm(y[t] ~ x[t] + y[t-1])
stargazer(reg1, reg2, type = "html")


	Dependent variable:

	y	y[t]
	(1)	(2)

x	0.818^***
	(0.074)

x[t]		0.083^*
		(0.043)

y[t - 1]		0.905^***
		(0.038)

Constant	2.493^***	0.238^*
	(0.221)	(0.132)


Observations	120	119
R²	0.511	0.918
Adjusted R²	0.506	0.916
Residual Std. Error	2.424 (df = 118)	1.000 (df = 116)
F Statistic	123.086^*** (df = 1; 118)	647.680^*** (df = 2; 116)

Note:	p<0.1; p<0.05; p<0.01

Spurious Regression

Warnings:

Check stationarity of the residuals
- Regression is spurious if residuals have a unit root.
- tseries::adf.test(lm(y ~ x)$residuals)
Results change dramatically if lagged $y$ is included
If spurious, make stationary!

Spurious Regression

Code

adf.test(reg1$residuals)
adf.test(reg2$residuals)


    Augmented Dickey-Fuller Test

data:  reg1$residuals
Dickey-Fuller = -3.0513, Lag order = 4, p-value = 0.1399
alternative hypothesis: stationary


    Augmented Dickey-Fuller Test

data:  reg2$residuals
Dickey-Fuller = -3.9078, Lag order = 4, p-value = 0.01613
alternative hypothesis: stationary

Integration

Time series with unit roots are said to be integrated or $I(1)$

$I(1)$ is unit root
$I(0)$ is stationary
$I(d)$ is $d$’th difference stationary
- Hansen says $I(2)$ series are price levels and money supply

Money Supply

Cointegration

Two series are cointegrated if a linear combination of them has a lower level of integration.

If $Y$ and $X$ are $I(1)$, but $Y - \theta X$ is $I(0)$, they are cointegrated.
Also consider two proportional time series.
- If $\frac{Y}{X}$ is stationary, then $log(Y) - \theta log(X)$ is $I(0)$

Another way to think about it: two series are cointegrated if there is a (relatively) constant distance between the two.

Cointegration

Cointegration

Again, $y$ and $x$ are coinegrated if:

\[z_t = y_t - \theta x_t \ \ \text{is} \ \ I(0)\]

$\theta$ is sometimes known from theory

In the case of term spread, it is $1$.
This can also be estimated.

Cointegration

Code

pi <- read.csv("../data/inc_pce.csv")
pi$DATE <- ymd(pi$DATE)
par(mfrow = c(1, 2))
plot(pi$DATE, pi$PI, type = "l",
     ylim = range(pi$PI, pi$PCE),
     xlab = "Date", ylab = "log($)",
     col = "tomato")
lines(pi$DATE, pi$PCE, type = "l", col = "dodgerblue")
legend("topleft", legend = c("PI", "PCE"),
       col = c("tomato", "dodgerblue"),
       lty = 1, bty = "n")

plot(pi$DATE, log(pi$PI), type = "l",
     ylim = log(range(pi$PI, pi$PCE)),
     xlab = "Date", ylab = "log($)",
     col = "tomato")
lines(pi$DATE, log(pi$PCE), type = "l", col = "dodgerblue")
legend("topleft", legend = c("PI", "PCE"),
       col = c("tomato", "dodgerblue"),
       lty = 1, bty = "n")

Cointegration

Code

reg <- lm(log(PCE) ~ log(PI), data = pi)
summary(reg)


Call:
lm(formula = log(PCE) ~ log(PI), data = pi)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.059112 -0.014470 -0.003781  0.016900  0.057097 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.3604532  0.0056156  -64.19   <2e-16 ***
log(PI)      1.0134213  0.0006801 1490.18   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.02196 on 730 degrees of freedom
Multiple R-squared:  0.9997,    Adjusted R-squared:  0.9997 
F-statistic: 2.221e+06 on 1 and 730 DF,  p-value: < 2.2e-16

Cointegration

Code

plot(pi$DATE,
     reg$residuals,
     type = "l")

Cointegration

Code

tseries::adf.test(reg$residuals)


    Augmented Dickey-Fuller Test

data:  reg$residuals
Dickey-Fuller = -2.9276, Lag order = 9, p-value = 0.1857
alternative hypothesis: stationary

Cointegration

Does the comovement of the 10-Year and 3-Month indicate a spurious relationship or cointegration?

They are both certainly non-stationary.

Cointegration

Code

fred <- read.csv("../data/spread.csv")
fred$DATE <- ymd(fred$DATE)
fred <- fred[fred$DGS10 != ".",]
fred <- fred[fred$DATE < ymd("2014-05-01"),]
colnames(fred) <- c("DATE", "t3m", "spread", "t120m")

for(i in 2:4) fred[,i] <- as.numeric(fred[,i])

reg <- lm(t120m ~ t3m, data = fred)
summary(reg)


Call:
lm(formula = t120m ~ t3m, data = fred)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.1886 -0.9453 -0.0241  0.9226  3.5382 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  2.48633    0.08521   29.18   <2e-16 ***
t3m          0.80997    0.01450   55.86   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.123 on 626 degrees of freedom
Multiple R-squared:  0.8329,    Adjusted R-squared:  0.8326 
F-statistic:  3120 on 1 and 626 DF,  p-value: < 2.2e-16

Cointegration

The residuals from this regression are $z_t$!

Code

plot(fred$DATE, reg$residuals, type = "l",
     xlab = "", ylab = "")

Cointegration

Code

tseries::adf.test(reg$residuals)


    Augmented Dickey-Fuller Test

data:  reg$residuals
Dickey-Fuller = -3.7392, Lag order = 8, p-value = 0.02204
alternative hypothesis: stationary

Cointegration

What does this have to do with spurious regression?

Stationary: use raw series
Non-Stationary: use differenced series
Non-Stationary & Cointegrated: use differenced series $+$ error correction

Cointegration

Error Correction Model: the change of one series is explained in terms of:

the lag of the difference between the series
lags of the differences of each series

\[\begin{align}\Delta y_t &= \Sigma_{i = 1}^{p} \alpha_{i} \Delta y_{t-i} \\ &+ \Sigma_{j = 1}^{q} \beta_{j} \Delta x_{t-j} \\ &+ \gamma z_{t-1} + \mu + e_t\end{align}\]

Cointegration

$z_t$ is the residual from the $Y$ on $X$ regression.

\[Y_t = \alpha + \beta X_t + z_t\]

If $z_t$ is stationary, then $Y$ and $X$ are cointegrated.

Cointegration

Code

t <- 2:nrow(fred)
d_t3m <- fred$t3m[t] - fred$t3m[t-1]
d_t120m <- fred$t120m[t] - fred$t120m[t-1]

t <- 2:length(d_t3m)
d_t3m_lag <- d_t3m[t-1]
d_t120m_lag <- d_t120m[t-1]
d_t3m <- d_t3m[t]
d_t120m <- d_t120m[t]

resid <- reg$residuals[2:(nrow(fred)-1)]
summary(lm(d_t120m ~ d_t120m_lag + d_t3m_lag + resid))


Call:
lm(formula = d_t120m ~ d_t120m_lag + d_t3m_lag + resid)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.7033 -0.1464 -0.0163  0.1495  1.4264 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.001580   0.011021  -0.143  0.88607    
d_t120m_lag  0.346914   0.046645   7.437 3.42e-13 ***
d_t3m_lag   -0.051878   0.030530  -1.699  0.08977 .  
resid       -0.029292   0.009943  -2.946  0.00334 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.2757 on 622 degrees of freedom
Multiple R-squared:  0.1081,    Adjusted R-squared:  0.1038 
F-statistic: 25.12 on 3 and 622 DF,  p-value: 2.364e-15